Theorem pwpwuni 44997

Description: Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
pwpwuni	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))

Proof of Theorem pwpwuni

Step	Hyp	Ref	Expression
1		elpwg 4608	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝐴 ⊆ 𝒫 𝐵))
2		sspwuni 5105	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
4		uniexg 7759	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
5		elpwg 4608	. . . 4 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
6	4, 5	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∈ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵))
7	6	bicomd 223	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ⊆ 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))
8	1, 3, 7	3bitrd 305	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605  ∪ cuni 4912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-ss 3980  df-pw 4607  df-uni 4913

This theorem is referenced by:  psmeasurelem  46426